The ratio of incomes of two persons A and B is `3:4` and the ratio of their expenditures is `5:7` . If their savings are Rs.15,000 annually find their annual incomes.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables for Incomes
Let the incomes of A and B be represented as:
- Income of A = 3x
- Income of B = 4x

### Step 2: Define Variables for Expenditures
Let the expenditures of A and B be represented as:
- Expenditure of A = 5y
- Expenditure of B = 7y

### Step 3: Set Up the Savings Equation
According to the problem, the savings of A and B can be expressed as:
- Savings of A = Income of A - Expenditure of A = 3x - 5y
- Savings of B = Income of B - Expenditure of B = 4x - 7y

Since their total savings are Rs. 15,000, we can write:
\[ (3x - 5y) + (4x - 7y) = 15,000 \]
This simplifies to:
\[ 7x - 12y = 15,000 \quad \text{(Equation 1)} \]

### Step 4: Express Expenditures in Terms of Incomes
From the ratios of expenditures, we know:
\[ \frac{5y}{7y} = \frac{3x}{4x} \]
This implies:
\[ \frac{5}{7} = \frac{3x}{4x} \]
Cross-multiplying gives:
\[ 5 \cdot 4x = 7 \cdot 3x \]
This simplifies to:
\[ 20x = 21x \]
Rearranging gives:
\[ 20x - 21x = 0 \quad \Rightarrow \quad x = 0 \]
This is incorrect, so we will use the savings equation directly.

### Step 5: Substitute Expenditures into the Savings Equation
From the ratio of expenditures, we can express y in terms of x:
\[ y = \frac{7}{5}(3x) \quad \Rightarrow \quad y = \frac{21x}{5} \]

### Step 6: Substitute y into Equation 1
Substituting y into Equation 1:
\[ 7x - 12\left(\frac{21x}{5}\right) = 15,000 \]
Multiply through by 5 to eliminate the fraction:
\[ 35x - 12 \cdot 21x = 75,000 \]
This simplifies to:
\[ 35x - 252x = 75,000 \]
Combining like terms gives:
\[ -217x = 75,000 \]
Thus,
\[ x = \frac{75,000}{217} \approx 345.65 \]

### Step 7: Calculate Incomes
Now we can find the incomes:
- Income of A = 3x = \(3 \times 345.65 \approx 1036.95\)
- Income of B = 4x = \(4 \times 345.65 \approx 1382.60\)

### Step 8: Final Answer
Thus, the annual incomes of A and B are approximately:
- Income of A = Rs. 1,036.95
- Income of B = Rs. 1,382.60